3d cursor x,y,z does not move with mesh
This behaviour is absolutely fine: the bones move vertices only, which have some weight related to their respective vertex group (with the same name). The object (its origin) stays still. If you move the object in object mode you will have weird issues unless you move the rig accordingly.

The usual way to animate is setting both object and rig at the center of the world in object mode, then in pose mode use the root bone to grab your character where he needs to be, and start the animation from there. The root bone can stay still while the rest of the bones are animating the mesh, or can be moved along a path in case of walk cycles or similar.